The Complexity of Combinatorial Optimization Problems on $d$-Dimensional Boxes

The MAXIMUM INDEPENDENT SET problem in $d$-box graphs, i.e., in intersection graphs of axis-parallel rectangles in $\mathbb{R}^d$, is known to be NP-hard for any fixed $d\geq 2$. A challenging open problem is that of how closely the solution can be approximated by a polynomial time algorithm. For the restricted case of $d$-boxes with bounded aspect ratio a PTAS exists [T. Erlebach, K. Jansen, and E. Seidel, SIAM J. Comput., 34 (2005), pp. 1302-1323]. In the general case no polynomial time algorithm with approximation ratio $o(\log^{d-1} n)$ for a set of $n$ $d$-boxes is known. In this paper we prove APX-hardness of the MAXIMUM INDEPENDENT SET problem in $d$-box graphs for any fixed $d\geq 3$. We give an explicit lower bound $\frac{245}{244}$ on efficient approximability for this problem unless $\PP=\text{\rm NP}$. Additionally, we provide a generic method how to prove APX-hardness for other graph optimization problems in $d$-box graphs for any fixed $d\geq 3$.